Optimal. Leaf size=80 \[ -\frac{1}{9} \left (3 x^2+5 x+2\right )^{3/2}+\frac{35}{72} (6 x+5) \sqrt{3 x^2+5 x+2}-\frac{35 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{144 \sqrt{3}} \]
[Out]
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Rubi [A] time = 0.0552115, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{1}{9} \left (3 x^2+5 x+2\right )^{3/2}+\frac{35}{72} (6 x+5) \sqrt{3 x^2+5 x+2}-\frac{35 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{144 \sqrt{3}} \]
Antiderivative was successfully verified.
[In] Int[(5 - x)*Sqrt[2 + 5*x + 3*x^2],x]
[Out]
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Rubi in Sympy [A] time = 6.20666, size = 71, normalized size = 0.89 \[ \frac{35 \left (6 x + 5\right ) \sqrt{3 x^{2} + 5 x + 2}}{72} - \frac{\left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{9} - \frac{35 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 5\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{432} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((5-x)*(3*x**2+5*x+2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.039914, size = 60, normalized size = 0.75 \[ \frac{1}{432} \left (-6 \sqrt{3 x^2+5 x+2} \left (24 x^2-170 x-159\right )-35 \sqrt{3} \log \left (2 \sqrt{9 x^2+15 x+6}+6 x+5\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(5 - x)*Sqrt[2 + 5*x + 3*x^2],x]
[Out]
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Maple [A] time = 0.007, size = 64, normalized size = 0.8 \[ -{\frac{1}{9} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}+{\frac{175+210\,x}{72}\sqrt{3\,{x}^{2}+5\,x+2}}-{\frac{35\,\sqrt{3}}{432}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((5-x)*(3*x^2+5*x+2)^(1/2),x)
[Out]
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Maxima [A] time = 0.769649, size = 97, normalized size = 1.21 \[ -\frac{1}{9} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} + \frac{35}{12} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x - \frac{35}{432} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac{175}{72} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(3*x^2 + 5*x + 2)*(x - 5),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.273507, size = 95, normalized size = 1.19 \[ -\frac{1}{864} \, \sqrt{3}{\left (4 \, \sqrt{3}{\left (24 \, x^{2} - 170 \, x - 159\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - 35 \, \log \left (\sqrt{3}{\left (72 \, x^{2} + 120 \, x + 49\right )} - 12 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(3*x^2 + 5*x + 2)*(x - 5),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \int x \sqrt{3 x^{2} + 5 x + 2}\, dx - \int \left (- 5 \sqrt{3 x^{2} + 5 x + 2}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5-x)*(3*x**2+5*x+2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.266799, size = 80, normalized size = 1. \[ -\frac{1}{72} \,{\left (2 \,{\left (12 \, x - 85\right )} x - 159\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{35}{432} \, \sqrt{3}{\rm ln}\left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(3*x^2 + 5*x + 2)*(x - 5),x, algorithm="giac")
[Out]